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Oblique asymptotes - What am I doing wrong?

  1. Dec 26, 2007 #1
    f(x) = (2x^3 + 4x^2 - x + 1) / (-x^2 - x + 2)

    The limit of this function as x approaches infinity is the oblique asymptote f(x) = -2x - 2
    This can be verified by performing long division with the two polynomials to get:

    f(x) = -2x -2 + (x +5)/(-x^2 - x + 2)
    as x -> infinity, the term (x +5)/(-x^2 - x + 2) -> zero

    Now my question is why does the following not work:

    f(x) = (2x^3 + 4x^2 - x + 1)/(-x^2 - x + 2) * (1/x^2)/(1/x^2)

    f(x) = (2x + 4 - 1/x + 1/x^2) / (-1 - 1/x + 2/x^2)

    lim f(x) as x -> infinity should then be -2x - 4 because all other terms approach zero as x gets larger and larger right?

    why is this wrong? I am so confused... I thought the answers should have been the same!!!
     
  2. jcsd
  3. Dec 26, 2007 #2

    mathman

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    Gold Member

    The denominator in your expression makes a contribution to the constant term.
    1/(-1-1/x)~-1+1/x. Therefore f(x)~(2x+4)(-1+1/x)~(-2x-4)+(2+4/x)~-2x-2
     
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